It is well known that the convergents of a continued fraction (i.e., the fraction you obtain by truncating a continued fraction at some point) are the best rational approximations for the number. The first ten convergents of are

, , , , , , , , , .

Notice that , so . Thus . This was John D. Cook’s observation.

Similarly, , so , and . The other two approximations come from the next two convergents.

After the conversation on Twitter, I started playing a little more.

First, I noticed that

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This comes from the third convergent. Since implies that , .

Actually, we can do better than that:

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The approximation, implies that , so .

Next, I noticed that

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This comes from , which implies that . Thus

Finally, I noticed that

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In this case, gives , and . We obtain the result by substituting the previous approximation for .

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One footnote: Good approximations to pi lead to great approximations to 1 when you take sines. Taylor series says sin(pi/2 + h) is approximately 1 – 0.5 h^2 for small h. So the error in the approximation for pi gets squared.